On the number of matroids
N. Bansal, R.A. Pendavingh, J.G. van der Pol

TL;DR
This paper tightens bounds on the number of matroids on n elements, showing that the logarithm of the logarithm of their count is close to Knuth's lower bound, using structural and combinatorial properties.
Contribution
It proves an upper bound on the number of matroids that closely matches Knuth's lower bound, refining the asymptotic estimate.
Findings
Established an upper bound within an additive term of Knuth's lower bound.
Used structural properties of non-bases and Johnson graph to represent matroids efficiently.
Confirmed that the number of matroids grows roughly as suggested by earlier bounds.
Abstract
We consider the problem of determining , the number of matroids on elements. The best known lower bound on is due to Knuth (1974) who showed that is at least . On the other hand, Piff (1973) showed that , and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on that is within an additive term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
On the Number of Matroids· youtube
Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Complexity and Algorithms in Graphs
